A Method for Investigating the Nonlinear Bends of the Cylinders of a Web Offset Printing Station

Metoda raziskave nelinearnih upogibov valjev pri rotacijskem

ofsetnem tiskarskem stroju

A Method for Investigating the Nonlinear Bends of the Cylinders of a Web Offset

Printing Station

Vytautas Kazimieras Augustaitis - Nikolaj Šešok - Igor Iljin (Vilnius Gediminas Technical University, Lithuania)

Elastièni pomiki valjev s plošè, valjev z gumijasto oblogo in tiskovnih valjev ter še posebej upogibov valjev, ki se pojavijo med vrtenjem valjev, pritiskajo drug ob drugega prek tanke elastiène gumirane tkanine, ki pokriva valja z gumijasto oblogo (valji so namešèeni tako, da je v vsakem paru valjev vsaj en valj z gumijasto oblogo), moèno vplivajo na delovanje rotacijskega ofsetnega tiskarskega stroja. Valji zaènejo pritiskati drug ob drugega zaradi pomikanja njihovih ležajnih sklopov.

Prispevek opisuje metodo raziskovanja upogibov valjev in drugih elastiènih pomikov, pa tudi sprememb tlaka vzdolž valjev, ki pritiskajo drug ob drugega. Med raziskavo smo ocenili nelinearne znaèilnosti elastiènosti sklopov ležajev, sile teže in lege osi vrtenja valjev glede na njihove medsebojne vplive. Obravnavani problem smo rešili kot zahteven nelinearni statièni problem, pri katerem smo na robovih delovnih površin valjev, ki pritiskajo drug ob drugega, poznali potrebne vrednosti tlaka, pri tem pa so bili pomiki sklopov ležajev, ki so potrebni za nastanek omenjenih vrednosti tlaka, na zaèetku neznani.

Z uporabo metode na specifiènem primeru smo pokazali, da so upogibi valjev bistveno vplivali na porazdelitev tlaka vzdolž dotikalnega predela valjev sodobnega tiskarskega stroja in s tem tudi na kakovost tiska.

© 2006 Strojniški vestnik. Vse pravice pridržane.

(Kljuène besede: rotacijsko ofsetno tiskanje, stroji tiskarski, valji, upogibanje, metode raziskovalne)

The elastic shifts of the plate, blanket and impression cylinders and, initially, the bends of the cylinders that appear during the rotation of the cylinders that are pressed against each other via the thin elastic cloth (blanket) that covers the blanket cylinders (the cylinders are located in such way that at least one blanket cylinder exists in each pair of cylinders) have a considerable impact on the operation of web offset printing stations. The cylinders are compressed by shifting their bearing assemblies.

The method of investigating the mentioned bends of the cylinders and other elastic shifts as well as the changes of the pressure along the cylinders that are pressed against each other are described. The nonlinear features of the elasticity of the assemblies of the cylinder bearings, the force of gravity and the location of the axes of the rotation of the cylinders with respect to each other are assessed. The problem under discussion is settled as a complex nonlinear static problem, where the required pressures at the edges of the working surfaces of the cylinders that are pressed against each other are provided and the shifts of the assemblies of the cylinder bearings that are required to ensure the said values of the pressures are unknown at the start.

On the application of the method to the specified example, it was shown that the bends of the cylinders noticeably impact on the distribution of the pressure along the contact zone of the cylinders in a modern printing press and on the quality of the prints.

© 2006 Journal of Mechanical Engineering. All rights reserved.

(Keywords: web offset printing, printing press, cylinders, bending, investigation methods)

0 UVOD

V tiskarski industriji je moèno razširjena raba rotacijskih ofsetnih strojev. Najpomembnejši del takšnega stroja je njegova tiskarska enota ([1] in [2]).

0 INTRODUCTION

Najpomembnejše komponente tiskarske enote vkljuèujejo valja s plošèo, valja z gumijasto oblogo in (v nekaterih primerih) tiskovna valja, ki sta vpeta v kotalne ležaje v osrednjem delu stroja. Mehanizma za dovajanje barve in vlaženje sta prav tako pomembna, vendar o njiju tu ne bomo govorili. Površini valjev z gumijasto oblogo sta prevleèeni s posebno tanko elastièno tesnilko (oblogo), ki ima gladko zunanjo površino. Vsi valji pritiskajo drug ob drugega po celotni dolžini in so v tiskarski enoti namešèeni tako, da ima v vsakem paru priležnih valjev vsaj en valj gumijasto oblogo. Zato pa valji pritiskajo drug ob drugega v smeri deformacije gumijaste obloge. V manjši meri lahko tlak krmilimo s spreminajnjem razdalje med osmi vrtenja valjev. To dosežemo z vrtenjem valjènih ležajev, ki so montirani v izsrednih pušah. Med delovanjem tiskarske enote, ko se vsi valji (pritisnjeni drug ob drugega) vrtijo brez drsenja, se odtisi prenesejo iz navlaženih tiskarskih predlog, ki sta pritrjeni na valja za plošèo, na površino gumijaste obloge in od tam na neskonèni trak, ki se v primeru obojestranskega tiskanja premika med vrteèima se valjema z gumijasto oblogo (ki pritiskata drug ob drugega), v primeru enostranskega tiskanja pa med valjem z gumijasto oblogo in tiskovnim valjem (ki tudi pritiskata drug ob drugega).

Pritisk valjev drug ob drugega prek elastiène obloge povzroèa absolutne in relativne elastiène pomike valjev (te elastiène pomike kasneje imenujemo kar pomiki). Ti pomiki sestojijo iz preènih premih pomikov sklopov valjènih ležajev, upogibanja valjev in kotnih pomikov (odstopanj) njihovih prerezov.

Tema tega prispevka je iskanje rešitve problema slabše kakovosti delovanja tiskarskega stroja – razvoj metod za raèunalniško podprto analitièno raziskavo upogibanja obloge, plošèe in tiskovnih valjev, ki ga povzroèa pritiskanje valjev preko obloge. Ti upogibi imajo negativen uèinek na kakovost tiskanja, pa tudi na njegovo izrabo in njegove dinamiène znaèilnosti. Upogibi valjev so povezani z njihovimi pomiki in jih moramo zato raziskovati hkrati. Najveè pozornosti smo posvetili relativnim upogibom valjev, ker ti povzroèajo spremembe v tlaku med upognjenimi valji, takšne spremembe pa neposredno vplivajo na poslabšanje kakovosti tiskov.

Zdaj bomo namen prispevka opisali bolj podrobno. V strokovni literaturi nismo zasledili rešitev zgoraj opisanega problema, a je vendarle

The most important components of the printing station are the plate cylinders, the blanket cylinders and (in some cases) the impression cylinders, which are mounted in rolling bearings in the body of the equipment. The inking and moistening mechanisms are also important, but they are not discussed here. The surfaces of the blanket cylinders are coated with a special thin elastic gasket (blanket), the external surface of which is even. All the cylinders are pressed against each other along their generatrices and are positioned in the station in such a way that for any pair of adjacent cylinders at least one of them is a blanket cylinder. For this reason, all the cylinders are pressed against each other on a deformation of the blanket. The pressure is regulated in a narrow range by changing the distance between the axes of the cylinders’ rotation. This is achieved by rotating the bearings of the cylinders, which are mounted in eccentric hubs. During the operation of the printing press, when all the cylinders (pressed against each other) are rotating without sliding, the prints are transferred from the inked printing forms fixed to the plate cylinders to the surface of the blanket of the blanket cylinders, and from there to the paper web that moves between two rotating blanket cylinders (pressed against each other), in the case of double-side printing, and between the blanket cylinder and the impression cylinder (pressed against it) in the case of printing on one side of the paper.

Pressing the cylinders against each other via the elastic blanket causes absolute and relative elastic shifts of the cylinders (such elastic shifts are subsequently referred to as shifts). These shifts consist of transversal rectilinear shifts of the assemblies of cylinder bearings, the bending of the cylinders, and the angular shifts (deviations) of their cross-sections. The subject of this paper is the solution of the important problem of the quality of a printing station -the development of methods for a computer-aided analytical investigation of the bending of the blanket, the plate and the impression cylinders caused by their pressing against each other via the blanket. These bends adversely affect the quality of the printing as well as its exploitation and dynamic features. The cylinder bends are linked to their other shifts, so they should be investigated together. The most attention is paid to the relative bends of the cylinders, because they cause changes of pressure between the bent cylinders, and such changes directly reduce the quality of the prints.

nujno najti ustrezne rešitve. Na primer, metode ocenjevanja upogibanja valjev so navedene v virih [3] do [5]. Le-te lahko uporabimo pri razlagi vpliva tovrstnega upogibanja na postopek tiskanja, ne pa tudi pri oceni vpliva upogibanja valjev, ki so v posebnih delih tiskarskega stroja.

Da bi razložili metode raziskave upogibanja valjev, navedenih v tem prispevku, in ocenili pomembnost upogibanja, smo uporabili moèno razširjen rotacijski ofsetni tiskarski stroj za obojestransko tiskanje [2]; sestava njegovih valjev in shema njegove postavitve sta prikazani na sliki 1.

Obravnavana tiskarska postaja sestoji iz valjev s plošèo 1, 4 (s pritrjenima tiskarskima predlogama, ki tu nista prikazani), in valjev z gumijasto oblogo 2, 3, ki sta prevleèena z oblogama 5. Premeri valjev so oznaèeni z Dd (kjer je d=1, …, 4 –

zaporedna številka valja), njihove dolžine pa kot l1.

Valji imajo lahko odprtine z dolžino l3 in premerom dd.

Vsi valji pritiskajo drug ob drugega v smeri deformacije oblog 5 z dolžino l2; a1, a2, g

so koti osi vrtenja valjev. Valji se vrtijo v posebnih ležajih 7 z visoko togostjo in so montirani na stojalo 8 tiskarske enote ter so namešèeni na stožèastih vratovih s premeroma

F oziroma F*_{. Stiskanje oblog uravnavamo s}

references; however, there is an urgent need to find such solutions. For example, methods of assessing the bending of cylinders are provided in references [3] to [5]. These can be used to explain the impact of such bending on the printing process, but not for assessing the impact of bending cylinders located in specific parts of the printing press on the printing process.

For an explanation of the methods of investigating the cylinder bending described here and an assessment of the importance of this bending, a widely used double-sided web offset printing station [2] was used. The structures of the cylinders and a scheme of the arrangement are shown in Fig. 1 The printing station under discussion consists of two plate cylinders 1, 4 (with the printing forms fixed on them and not shown here) and two blanket cylinders 2, 3, coated with blankets 5. The diameters of the cylinders are marked as Dd (where d=1, …, 4 – the consecutive

number of a cylinder) and their lengths – as l1. The cylinders can have l3 long holes with the diameter dd.

When all the cylinders are pressed against each other on a deformation of l2 along the blankets 5, a1, a2,

g are the angles of the axes of rotation of the cylinders. The cylinders roll in special and precise high-rigidity bearings 7 that are mounted in the stand 8 of the printing station and put onto the conical necks of the cylinders with diameters F and F*_{, respectively. The compression}

Sl. 1. Shema postavitve valjev s plošèo in valjev z gumijasto oblogo v rotacijskem ofsetnem stroju za obojestransko tiskanje: a – lege osi vrtenja valjev O1–O1,..., O4–O4; b – postavitev valjev glede na CO2O3E

Fig.1. The scheme of the assembly of the plate and blanket cylinders in the web offset double-side printing station: a – the location of the axes of the rotation of cylinders O1–O1,..., O4–O4; b – the evolvent of the

cylinders according to CO2O3E 1

a

g

3 2 1

5 5

6

4

1

O

3

O

4

O

2

O C

E _a2

a

5

5

dd

D

L

2

l

1

l

4

O

3

O

2

O

1

O 7

8 1

O

3

O

4

O

2

O

3

l

F

F

*

s

b

x

l

y

l ps

ps

pm y

2

l p

O

m

b

x

px premikanjem sklopov ležajev valja 2 v dveh

pravokotnih smereh ter premikanjem ležajev valja 4 v eni smeri (na sliki 1 so smeri krmiljenja prikazane s pušèicami). Med pritiskanjem se valji (ki jih prek zobnikov obraèa elektrièni motor) vrtijo, ne da bi drgnili ob delovne površine, pri tem pa se med valjema z gumijasto oblogo 2 in 3 pomika papirni trak 6. Tiskanje poteka na obeh straneh traku. Delovna površina valja je tisti del površine, ki je prevleèen z oblogo (pri valjih z gumijasto oblogo) in del, ki je v stiku z oblogo (pri valjih za plošèo in tiskovnih valjih).

Ko valji pritiskajo drug ob drugega, se obloga deformira in med valji nastanejo pasovno oblikovani dotikalni predeli. V obravnavanem primeru (slika 1) obstajajo trije dotikalni predeli (j = 3): med valjema 1 in 2, 2 in 3 ter 3 in 4. Na katerikoli toèki j-tega stiènega predela je stiskanje obloge Dj. Najveèje stiskanje obloge Dm,j vzdolž j-tega dotikalnega predela (j=1, 2, 3) in v preèni smeri nastane približno v sredini omenjenega predela. Opazili smo majhen odmik Dm,j od sredine dotikalnega pasu in v smeri, ki je nasprotna smeri premih hitrosti površin vrteèih se valjev, ki pritiskajo drug ob drugega. ([4] in [5]). Tukaj, kakor tudi v primerih [3] do [5], tega odmika nismo upoštevali. Vrednost Dm,j se spreminja zaradi upogibanja valjev: vrh (Dm,j)max = Ds,j najdemo na robovih delovnih površin valjev, dol pa v sredini (v primeru, ko se valja upogibata v nasprotnih

of the blankets is regulated by a shifting of the assemblies of the bearings of cylinder 2 in two perpendicular directions and the shifting of cylinder 4 in one direction (Fig. 1, the directions of the regulation are shown by arrows). During compression, the cylinders (rotated by electric engine via tooth gears) are rolling without sliding against their working surfaces and draw the paper tape 6 between the blanket cylinders 2 and 3. The prints are made on both sides of the tape. The working surface of a cylinder is the part of its surface that is coated with a blanket (in blanket cylinders) or the part having contact with it (in the plate and impression cylinders).

When pressing the cylinders against each other, the blanket between them is deformed and band-shaped contact zones appear between them. In the case under discussion (Fig. 1), there are three contact zones (j = 3): between the cylinders 1 and 2, 2 and 3, 3 and 4. At any point of the j-th contact zone the compression of the blanket is Dj. The maximum compression Dm,j of the blanket

along the j–th band of a zone (j=1, 2, 3) in a radial direction is obtained approximately in the middle of the mentioned band. The small deviation Dm,j from the middle of the band

and in the direction opposite to the one of the linear speeds of the surfaces of rotating cylinders pressed against each other was observed ([4] and [5]). Here, as in [3] to [5], this deviation is not taken into account. The value of Dm,j

changes because of the bending of the cylinders: the maximum (Dm,j)max = Ds,j is found at the ends of the working

surfaces of the cylinders and the minimum is found in the middle (in the case of bending of the cylinders in opposite directions). The pressure pj between the cylinders depends

Sl. 2. Porazdelitev tlaka p v j-tem dotikalnem predelu (indeks “j” smo tu opustili); bs, ps– širina dotikalnega predela in tlak na njegovih robovih; bm, pm– sta enaka na razdalji ly od zaèetka dotikalnega predela; px – tlak na katerikoli toèki odseka bm na razdalji lx od toèke najveèjega tlaka pm

Fig. 2. The distribution of the pressure p in the j-th contact zone of the cylinders (the index “j” has been omitted here); bs, ps – the width of the contact zone and the pressure at its ends; bm, pm – the same in ly distance from the beginning of the zone; px – pressure in any point of the section bm in lx distance from

smereh). Tlak pj med valji je odvisen od stiskanja obloge Dj. Naèin porazdelitve tlaka pj po j-tem dotikalnem predelu je prikazan na sliki 2.

Na podlagi izmerjenih podatkov ([5] in [6]) vemo, da med tlakoma pj, pm,j, napetostima sj, sm,jv tlaèenem delu obloge in njenima stiskoma Dj, Dm,j obstaja naslednje razmerje:

E in c sta Youngov modul in nedeformirano debelino obloge. Nespremenljivi koeficient h = 1,2 do 1,5 za rotacijsko ofsetno tiskanje ([5] in [6]):

Nespremenljivi koeficiet h in obièajni Youngov modul obloge E sta uporabljena tako kakor v virih [4] do [6]. Da bi dobili vrhunski tisk, tlak pm,j ne sme prekoraèiti najveèje

( )

* , _max m j

p in

najmanjše

( )

pm j, *_min dovoljene omejitve. Ti dve

omejitvi ustrezata razliki med dovoljenima stiskoma obloge.

Vrednosti dovoljenih tlakov in stiskov so odvisne od vrste obloge, prièakovane kakovosti tiska in od drugih dejavnikov. Najpogosteje jih dobimo s preizkusi po doloèitvi naslednjih odvisnosti: tlak med valji in stisk obloge.

Èe uporabimo enaèbi po [4] in [5], sta h in E [6]:

Iz enaèbe (1) izpeljemo naslednje razmerje med pm,j in Dm,j :

on the compression of the blanket Dj. The character of the

distribution of pressure pj in the j–th contact zone is shown

in Fig. 2.

It is known from experimental data ([5] and [6]) that the following interrelation between the pressures

pj, pm,j, and the tensions sj, sm,j in the compressed

part of the blanket and its compressions Dj, Dm,j exists:

(1).

Here, E and c are the Young’s modulus and the non-deformed thickness of the blanket. The constant coefficient h=1.2 to 1.5 for offset web printing ([5] and [6]):

(2)

(3)

(4).

The constant coefficient h and the conventional Young’s modulus of the blanket E are set as in [4] to [6]. In order to obtain high-quality prints, the pressure pm,j

should not overstep the maximum

( )

pm j, *_max and

minimum

( )

pm j, *_min permissible limits. These limits

correspond to the difference between the permissible compressions of the blankets

(5).

The values of the permissible pressures and compressions depend on the type of blanket, the required quality of the prints and other factors. Most frequently, they are found in an experimental way after the formation of the following dependences: the pressure between the cylinders and the compression of the blanket.

Using equations from [4] and [5], the expressions for h and E [6] can be written as:

(6)

(7).

From (1), the interrelation between pm,j and

Dm,j is found to be:

(8).

,

, ,

; m j

j

j j m j m j

p E p E

c c

h _D h

D

s æ ö s æ ö

= = _{ç ÷} = = _{ç ÷}

è ø è ø

2 ,

, 2

4

1 x j

j m j j

l b

D =D æç_ç - ö÷_÷

è ø

2

, ₈j ; 2 2 , ( j = 1, 2, 3 )

m j j j m j

j

b

b B

B

D = = D

(

)

(

1

)

(

4

(

)

)

1 3 2 2 3

1 4

2 _; 2 _; 2 _;

2

2 2 2 2

D D c D D c D c

B B B D D D

D D c D D c

+ + +

= = = = =

+ + + +

( ) ( )

* * *

, _max , _min

m m j m j

d

D = D - D

(

) (

)

( )

( )

{

}

* *

, _max , _min

* _* *

, _min , _min

lg

lg

m j m j

m j m m j

p p

d

h

D D D

é ù

ê ú

ë û

=

é ₊ ù

ê ú

ë û

( )

*

( )

* _*

( ) ( )

* *

, _max , _min , _min , _min

m j m j m m j m j

E c p₌ h é _{l +D}d ù ₌c ph _D

ê ú

ë û

1/ , , m j m j

p c E

h

D _{= ç}æ ö÷

Porazdelitev tlaka p po j-tem dotikalnem predelu je prikazana na sliki 2.

Tlak pm,j je doloèen kot prvi pogoj. Da bi zagotovili njegovo potrebno vrednost, morajo valji pritiskati drug ob drugega z moèjo, ki povzroèa potrebno vrednost najveèjega stiska obloge Dm,j. Zaželeno bi bilo, da je pm,j= konst, vendar to ni mogoèe zaradi upogibanja valjev. Potrebni pm,j (kakor tudi Dm,j) je lahko povpreèna vrednost tlaka pm,j oziroma njegova vrednost na kateremkoli prerezu valjev, na primer, znotraj valja ali na robovih delovnih površin. V tem prispevku predpostavljamo, da sta potrebni najveèji tlak med valji in ustrezni najveèji stisk obloge doloèena na robovih delovnih površin valjev, torej je tlak ps,j stisk obloge Ds,j.

Stisk obloge povzroèajo pomiki sklopov ležajev; ti pomiki se ne ujemajo z Ds,j, ker med pomikanjem sklopi ležajev in robovi valjev postanejo deformirani. Soodnos med pomiki sklopov ležajev in Ds,j za katerokoli dvojico valjev, ki pritiskata drug ob drugega, dobimo z izraèunom pomikov vseh valjev tiskarske enote, ko ocenimo nelinearnost odziva elastiènosti obloge.

0.1 Cilj raziskave

Cilj prispevka je razvoj analitiènih, raèunalniško podprtih matematiènih metod, ki jih lahko uporabimo za izraèun in raziskavo zakonitosti deformacij valjev s plošèo, valjev z gumijasto oblogo in tiskovnih valjev (pri rotacijskem tiskarskem stroju), ki pritiskajo drug ob drugega prek obloge, pa tudi tlaka na dotikalnih predelih valjev. Omen-jene metode prav tako omogoèijo doloèitev vpliva deformacije valjev na analizo postopka tiskanja in doloèitev vrednosti pomikov sklopov ležajev, ki povzroèijo specifiène vrednosti stiska obloge Ds,j in tlaka ps,j.

Rezultate raziskave lahko uporabimo pri oblikovanju in izboljšavi tiskarskih enot in pri oceni kakovosti novih tiskarskih strojev še preden jih zaènemo uporabljati.

Razvijanje omenjenih metod je bilo zahtevno, ker je odnos med stiskalnimi silami valjev in stiskom oblog, ki jih povzroèijo valji, nelinearen in ker pomiki ležajev, ki omogoèijo potrebno stiskanje oblog Ds,j, niso vnaprej znani. Vendar pa je nujno potrebno, da ugotovimo

The distribution of the pressure p in the j-th contact zone is shown in Fig. 2.

The pressure pm,j is set in the requirements. In

order to ensure the required value of the pressure, the cylinders are pressed against each other with a force that causes the required value of the maximum compression

Dm,j of the blanket. It is desirable for pm,j= const, but this is

impossible because of the bending of the cylinders. The required pm,j (as well as Dm,j) can be considered as the

average value of the pressure pm,j or its value in any

cross-section of the cylinders, for example, inside the cylinders or at the edges of their working surfaces. In this paper, it is considered that the required maximum pressure between the cylinders and the corresponding maximum compression of the blanket are determined at the edges of the working surfaces of the cylinders, i.e., they are the pressures ps,j and the blanket’s compressions are Ds,j.

The compression of the blanket is caused by shifts of the assemblies of cylinder bearings; these shifts do not coincide with Ds,j, because the assemblies of bearings and

the ends of the cylinders are deformed during the shifting. The interrelation between the shifts of the assemblies of bearings and Ds,j for any pair of cylinders that are pressed

against each other is found by calculating the shifts of all the cylinders of the printing station when the non-linearity of the response of the blanket’s elasticity is assessed.

0.1 Object of the Paper

The object of the paper is to develop analytical computer-aided calculation methods that are usable for the calculation and investigation of the regularities of deformations of the plate, blanket and impression cylinders (in the web printing press) that are pressed against each other via the blanket as well as the pressure on the contact zones of the cylinders; a determination of the impact of the deformation of the cylinders on an investigation of the printing process; a determination of the values of shifts of the bearings assemblies of such cylinders that cause the set values of blanket compression Ds,j and pressure ps,j.

The results of the paper may be applied to designing and improving printing stations and an assessment of the quality of the acquired printing presses before starting their exploitation.

njihove vrednosti, kajti, kakor bomo pokazali v tem prispevku, je prav zaradi tega problema naša raziskava drugaèna od obièajnih iskanj rešitve sistema nelinearnih enaèb.

1 SHEMA IZRAÈUNOV

Shema izraèunov, ki se nanaša na tiskarsko enoto, predstavljeno na sliki 1, je prikazana na sliki 3. Shema kaže, da so absolutni pomiki valjev izraèunani, njihove medsebojne razlike pa lahko uporabimo za doloèitev relativnih pomikov valjev in s tem vrednosti

Dm,j, Ds,j, pm,j, ps,j, ki nas najbolj zanimajo.

V shemi izraèunov je bila elastiènost obloge in ležajev ocenjena z uporabo diskretnih elastiènih elementov (njihovi odzivi so opisani z ustreznimi koeficienti togosti) in z uporabo metode konènih elementov, ki izloèi pomike valjev. Analogne sheme izraèunov smo oblikovali tudi za sklope valjev, ki so drugaèe razporejeni in so sestavljeni iz drugaènega števila posameznih valjev.

Valji so razstavljeni na plošèate konène elemente, ki so oblikovani kot valji in prisekani stožci (slika 3, c, d). Z uporabo enaèb smo ocenili upogibe in druge pomike valjev v ravninah, ki potekajo skozi naslednje teoretiène osi vrtenja nedeformiranih valjev (te osi kasneje imenujemo osi vrtenja valjev): valj 1 – ravnina osi vrtenja valja 1 in valja 2 je skladna z osjo koordinat x1;

valj 4 – ravnina osi vrtenja valjev 4 in 3 je skladna z osema koordinat x2, x3 in dve ravnini, ki ležita

pravokotno na to ravnino – ena gre zkozi os vrtenja valja 2, druga pa skozi os vrtenja valja 3 (prva ravnina ima os koordinat y2 in druga ima os

koordinat y3).

Tako so absolutni pomiki valja 1, in tudi valja 4 ocenjeni v eni ravnini (pomiki valja 1 v smeri osi koordinat x1 in pomiki valja 4 v smeri osi x4), pomiki

valjev 2 in 3 pa v dveh pravokotnih ravninah (pomiki valja 2 v smeri osi koordinat x2, y2 in pomiki valja 3 v

smeri osi koordinat x3, y3).

Predpostavljamo, da ima tiskarska enota simetrièno ravnino, ki poteka skozi središène toèke valjev (toèke O1,...,O4) in leži pravokotno na njihove

osi vrtenja (sl. 3, b). Zato so vse vrednosti, ki opisujejo valje, njihove pomike in sile, ki delujejo na valje v obeh polovicah valjev, ki jih deli ravnina, enake.

known in advance. It is necessary to find them and this, as we will show here, causes a difference in the problem under discussion from the usual problem of the solution of a system of non-linear equations.

1 THE SCHEME OF CALCULATIONS

The scheme of calculations, corresponding to the printing station, presented in Fig. 1, is shown in Fig. 3. According to this scheme the absolute shifts of the cylinders are calculated and the corresponding differences are used to find the relative shifts of the cylinders and the values Dm,j, Ds,j, pm,j, ps,j that are our main interest.

In the scheme of the calculations the elasticity of the blanket and the cylinder bearings was assessed by using discrete elastic elements (their responses are described with the corresponding coefficients of stiffness) and by applying the use of a method of finite elements that discretized shifts of the cylinders. Analogous schemes of the calculation are formed for other layouts and numbers of cylinders.

The cylinders are disintegrated to plane finite elements shaped as cylinders and truncated cones (Fig. 3, c, d). Using the equations, the bends and other shifts of the cylinders are assessed in planes, passing through the following theoretical axes of rotation of non-deformed cylinders (these axes are subsequently referred to as the axes of rotation of the cylinders): cylinder 1 – the plane of the axes of rotation of it and cylinder 2 according to the axis of the coordinates x1; cylinder 4 - the plane of the axes of rotation of cylinders 4 and 3 according to the axes of the coordinates x2, x3 and in two planes perpendicular to this plane – one of them goes through the axis of rotation of cylinder 2 and the other - through the axis of rotation of cylinder 3 (the first plane includes the axis of coordinates

y2 and the second plane – the axis of coordinates y3). Thus, the absolute shifts of cylinder 1, like that of cylinder 4, are assessed in one plane (the shifts of cylinder 1 in the direction of the axis of coordinates x1, and the shifts of cylinder 4 in the direction of the axis x4) and the shifts of cylinders 2 and 3, in two perpendicular planes (the shifts of cylinder 2 in the direction of the axes of coordinates x2, y2 and the shifts of cylinder 3 in the direction of the axes of coordinates x3, y3).

Vrednosti stiska oblog Ds,j (kadar so znane, so vrednosti tlaka ps,j izraèunane z enaèbo (1)) dobimo z uporabo nadzornih toèk A1,...,A4 v enotah konènih

elementov, ki so na levem robu delovnih površin valjev (sl. 3, b). Spremembe pri razdaljah A1A2, A2A3,

A3A4, ki jih dobimo z izraèunom, so enake Ds,j upoštevaje dovoljeno napako. Zaradi simetrije je dovolj, da ugotovimo navedene spremembe le na eni strani valja.

Ležaji valjev so v približku enaki linearnim elastiènim elementom, katerih koeficienti togosti kb,1,...,kb,4 so izraèunani tako kakor v viru [7] (sodobni

tiskarski stroji imajo valjaste kotalne ležaje, elastiènost njihovih sklopov pa je v približku premoèrtna).

Plast obloge med valjema smo simulirali z diskretnimi nelinearnimi elastiènimi elementi, ki povezujejo delovne površine dvojice valjev, ki pritiskajo drug ob drugega ob dotikališèih konènih

The values of the compression of the blankets Ds,j

(when they are known, the values of the pressure ps,j are

found according to Formula (1)) are found using the control points A1,...,A4 in the units of the finite elements, situated at the left-hand edge of the working surfaces of the cylinders (Fig. 3, b). Changes of the distances A1A2, A2A3,

A3A4, found in the calculation, are equal to Ds,j, with a

permissible error. Because of the symmetry, it is sufficient to find these changes at only one end of the cylinders.

The bearings of the cylinders are approximated with linear elastic elements whose coefficients of stiffness kb,1,...,kb,4 are calculated according to [7] (in modern printing stations, cylindrical rolling bearings are used, and the elasticity of their assemblies is approximately rectilinear).

The layer of blanket between the cylinders is simulated with discrete non-linear elastic elements that connect the working surfaces of pairs of cylinders that are pressed against each other at the junctions of the finite

Sl. 3. Shema izraèunov za tiskarski stroj, prikazan na sliki 1: a – pogled s strani; b – postavitev glede na osi vrtenja valjev; c, d – tipi konènih elementov, ki simulirajo valje; qM,...,qB, jM,...,jB– linearne in kotne

koordinate, ki doloèajo lege sklopov konènih elementov

Fig. 3. The scheme of calculations of the printing station shown in Fig. 1: a – the view from the end; b – the cylinders; qM,...,qB, jM,...,jB – the linear and angular coordinates that define the positions of the

assembles of the finite elements

a) b)

elementov (sl. 3, b). Te elemente definiramo s koeficienti togosti kj,v = k1,v; k2,v; k3,v, ki so odvisni od najveèjih vrednosti stiskanja obloge (n -zaporedno število dotikališè konènih elementov na oblogi (v primeru s sl. 3: n = 1, 2,..., 10) v j-tem dotikalnem pasu). Vsak od njih doloèi togost zv, dolgega valjastega izreza obloge. Vrednosti zv so

odvisne od dolžin ae konènih elementov na oblogi (sl. 3, b, c).

Po enaèbah (1) do (4) in (8) pa tudi virih ([4] do [6]) smo izpeljali naslednje izraze za koeficiente togosti elastiènih elementov, ki simulirajo oblogo:

Koeficient a1 = a3 = 1 ( j = 1, 3) in a2 = 0,5 ( j =

2). Glede na [4] in [5] dobimo naslednji izraz:

Torej so vsi koeficienti togosti kj,v funkcije vrednosti stiskanja obloge (Dm,j)v.

Papirni trak 6 se pomika med valjema z gumijasto oblogo 2 in 3 (sl. 1). Lahko bi dokazali, da ima elastiènost stisnjenega papirja zanemarljiv uèinek, zato je tu ne upoštevamo. Po potrebi bi elastiènost papirja lahko ocenili z ustreznim zmanjšanjem koeficientov togosti obloge med valjema 2 in 3.

Masa valjev lahko doseže nekaj sto kilogramov, zato jo moramo upoštevati. V shemi izraèunov (sl. 3, a, b) je masa valjev v približku enaka silam mase Gd,r (r = 1, 2, … – zaporedne številke dotikališè konènih elementov), ki delujejo na stikališèa konènih elementov.

Pri obravnavanem tiskarskem stroju se oblogi napneta zaradi pomikov sklopov ležajev drugega valja za razdalji (-dx2), dy2 in zaradi pomikov

sklopov ležajev èetrtega valja za razdaljo (-dx4) (sl.

3, a; vrednosti dx2 in dx4 sta negativni, ker pomikanje

poteka v smereh, ki sta nasprotni smerem pozitivnih smeri osi koordinat x2 in x4). To

dogajanje ustvarja naslednje sile pomikanja valjev 2 in 4, ki uèinkujejo na prikljuèke ležajev valjev (sl. 3, a, b):

Vrednosti pomikov dx2, dy2, dx4 in tudi sil Fx2,

Fy2, Fx4 so funkcije elastiènih pomikov valjev.

elements (Fig. 3, b). These elements are defined by the coefficients of stiffness, kj,v = k1,v; k2,v; k3,v, that depend on

the maximum values of the compression of the blanket (n

- the consecutive number of junctions of the finite elements covered by the blanket (in this case shown in Fig. 3: n= 1, 2,..., 10) in the j-th contact band). Each of them determines the stiffness of zv, the long cylindrical cut of the blanket.

The values of zv depend on the lengths ae of the finite

elements covered with the blanket (Fig. 3, b, c).

On the basis of the Formulas (1) to (4) and (8) as well as the references ([4] to [6]) we found the following expressions of the coefficients of stiffness of the elastic elements that simulate the blanket to be: (9). The coefficient a1 = a3 = 1 ( j = 1, 3) and a2 = 0.5 ( j = 2). According to [4] and [5]:

(10).

Therefore, all the coefficients of stiffness kj,v are

functions of the values of the compression of the blanket (Dm,j)v.

The paper tape 6 moves between the blanket cylinders 2 and 3 (Fig. 1). It can be shown that the elasticity of its compression has a negligible impact, so it is not taken into account here. If necessary, the elasticity of the paper can be assessed by the corresponding reduction of the coefficients of stiffness of the blanket between the cylinders 2 and 3.

The weight of the cylinders can reach some hundreds of kilograms, so it should be taken into account. In the scheme of the calculations (Fig. 3, a, b), the weight of the cylinders is approximated with the forces of gravity Gd,r (r = 1, 2, … - the consecutive

numbers of junctions of the finite elements) acting at the junctions of the finite elements.

In the printing station under discussion the blankets are stretched by shifting the assemblies of bearings of the second cylinder distances (-dx2), dy2 and the assemblies of the bearings of the fourth cylinder a distance (-dx4) (Fig. 3, a; the values dx2 anddx4 are negative, because the shifting is performed in directions opposite to the set positive directions of the axes of coordinates x2 and x4). This generates the following forces of the shifting of cylinders 2 and 4 that affect the cylinders in the fittings of their bearings (Fig. 3, a, b):

(11). The values of the shifts dx2, dy2, dx4 as well as of the forces Fx2, Fy2, Fx4 are functions of the elastic shifts of the cylinders.

( )

0,5

, 8 , ( j = 1, 2, 3)

j j j m j

k B E z ch h

n a ny D _n

-@

(

)

1 1 2 0,75 4 0,9375 4 0,4375 12

h h h

y@ + × + × + ×

2 ,2 2; 2 ,2 2; 4 ,4 4

x b x y b y x b x

2 ALGORITEM IZRAÈUNOV

2.1 Glavne znaèilnosti algoritma

Na kratko bomo opisali zaporedje izraèunov. Najprej smo ugotovili absolutne pomike valjev v smereh osi koordinat x1,...,x4, ki so

prikazani na sl. 3, a, b. Te pomike opišemo s posplošenimi Lagrangevimi koordinatami (v nadaljevanju jih imenujemo kar koordinate) q1,q2,...,qn (skupaj sestavljajo vektor q). Nato, z uporabo ustreznih razlik med omenjenimi koordinatami, relativne pomike med valji najdemo na ravninah, ki potekajo skozi osi vrtenja valjev 1 in 2, 3 in 4 (pomiki valjev 2 in 3 se preslikajo na te ravnine), ter skozi osi vrtenja valjev 2 in 3.

Vrednosti koordinat q lahko najdemo v rešitvi enaèb oblikovanih na njihovi osnovi. Èe so vrednosti pomikov sklopov ležajev dx2, dy2, dx4

znane, lahko omenjene koordinate dobimo z naslednjim nelinearnim sistemom algebraiènih enaèb:

Tu del elementov matrike togosti [K] vsebuje koeficiente tromosti gumijaste obloge kj,v, ki so funkcije vrednosti stiskanja obloge Dm,j pa tudi koordinat q.

Kljub temu pa je oblikovanje in reševanje enaèb tipa (12) zapleteno, ker so vrednosti stiska oblog Ds,j = Ds,j(q1,...,qn), ki so odvisne od vrednosti pomikov dx2, dy2, dx4, sicer znane, ne poznamo pa tudi

vrednosti samih dx2, dy2, dx4; prav tako je neznan

analitièni medsebojni odnos med temi vrednostmi. Zato so neznani tudi ustrezni izrazi komponent vektorja F.

Algoritem, ki je predstavljen spodaj, predlagamo kot mogoèo rešitev sistema enaèb tipa (12) z uporabo iterativne metode izraèuna, ki smo ga razvili prav za ta namen: z njegovo izpeljavo so linearizirani sistemi enaèb rešeni ob vsaki ponovitvi in oblikovanje nelinearnega sistema enaèb (12) sploh ni potrebno.

Izraèun poteka v veè stopnjah. Na vsaki stopnji smo preuèili naslednji linearni sistem algebraiènih enaèb, pridobljenih z lineariziranjem sistema (12) (metode lineariziranja so opisane v odstavku 2.2):

2 THE ALGORITHM OF THE CALCULATIONS

2.1 The Principal Features of the Algorithm

The sequence of the calculations will be briefly described. First of all, the absolute shifts of the cylinders in the directions of the axes of coordinates

x1,...,x4 shown in Fig. 3, a, b were found. These shifts are described by using the generalized Lagrange coordinates (subsequently, the coordinates) q1,q2,...,qn

(altogether they form the vector q). Then, by using the corresponding differences of the said coordinates, the relative shifts between the cylinders are found in the planes that pass through the axes of the rotation of cylinders 1 and 2, 3 and 4 (the shifts of cylinders 2 and 3 are projected onto these planes), and through the axes of rotation of cylinders 2 and 3.

The values of the coordinates of q can be found in the solution of the equations formed on their basis. If the values of the shifts dx2, dy2, dx4 of the assemblies of the cylinder bearings are known, the said coordinates can be found from the following non-linear system of algebraic equations:

(12). Here, a part of the elements of the matrix of stiffness [K] includes the coefficients of stiffness of the blanket kj,v

that are functions of the values of the compression Dm,j of

the blanket as well as of the coordinates of q.

However, the formation and solution of the equations of type (12) is complicated, because the values of the compression of the blankets Ds,j = Ds,j(q1,...,qn) that

depend on the values of the shifts dx2, dy2, dx4 are known, but not the values of dx2, dy2, dx4 themselves, and the analytical interrelation between these values is also unknown. So, the specific expressions of the components of the vector F are unknown as well.

The algorithm described below is proposed as a solution of the systems of (12)-type equations by using the iterative method of calculation that was developed for this purpose: on its realization, the linearized systems of equations are solved on each iteration and the formation of a non-linear system of Equations (12) is not required.

The calculation is performed in several stages. At each stage, the following rectilinear system of algebraic equations obtained on the linearization of the system (12) was examined (the methods of linearization are described in paragraph 2.2):

(13).

[

K( , , )q1K qn

]

q F=

(

d d dx2, y2, x4

)

+FG

( ) ( ) ( )

(

( ) ( ) ( )

)

2, 2, 4

i i i i i i

x y x G

d d d

é ù = +

Tu so elementi matrike togosti [K(i)_]

koeficienti togosti gumijaste obloge ( ) , i j

k n (njihov

izraèun je opisan v odstavku 2.2), pa tudi koeficienti togosti sklopov ležajev valjev, ki kažejo deformacije valjev. Vsi elementi matrike [K(i)_{] so}

nespremenljive.

Vektor sil F(i)_{, ki pomikajo valje v i-tem}

približku, je funkcija približne vrednosti pomikov sklopov ležajev ( )2, ( )2, ( )4

i i i x y x

d d d _{, ki smo jo dobili med}

izraèunom. Vsebuje le tri nespremenljive nenièelne dvojice komponent (dve enaki komponenti za vsak pomik):

Tudi komponente vektorja sil težnosti FG so nespremenljive.

Z uporabo raèunalniško podprte metode, ki sledi shemi izraèunov, smo oblikovali sistem enaèb (13), ki je prikazan na sliki 3. Za njegovo oblikovanje lahko uporabimo razliène algoritme, vendar pa priporoèamo naslednji algoritem, ki smo ga razvili. Najprej za posamezne valje oblikujemo enaèbe podsistema našega sistema (ne celotnega sistema); oblikujemo tudi elastiène elemente, ki simulirajo gumijasto oblogo in so umetno loèeni od enega valja. Tako v obravnavanem primeru sestavimo naslednje loèene sisteme enaèb šestih podsistemov: enaèbe valja 1 z elastiènimi elementi, ki simulirajo gumijasto oblogo in so loèeni od valja 2 (njihovi koeficienti togosti so ( )

1,i

k n)) – te enaèbe

opišejo pomike valja v smeri x1; enaèbe valja 2, ki

opišejo pomike valja v smeri x2, z elastiènimi

elementi, loèenimi od valja 3 (njihovi koeficienti togosti so ( )

2,i

k n); enaèbe valja 2 brez elastiènih

elementov, ki opišejo pomike valja v smeri y2; dva

analogna sistema enaèb brez elastiènih elementov, ki posamièno opišeta pomike valja 3 v smereh x3

in y3; enaèbe valja 4, ki opišejo pomike valja v

smeri x4 z elastiènimi elementi, loèenimi od valja 3

(njihovi koeficienti togosti so v). Poleg tega oblikujemo tudi linearne algebraiène povezovalne enaèbe, ki opišejo povezave prostih koncev elastiènih elementov; ti simulirajo gumijasto oblogo, pri èemer so valji umetno loèeni od elastiènih elementov.

Potem ko razvijemo vseh šest sistemov enaèb podsistemov ter povezovalne enaèbe, jih združimo v enoten sistem (13) z uporabo raèunalniško podprte metode. Takšna metoda oblikovanja enaèb, ki

Here, the elements of the matrix of stiffness [K(i)_{] are the coefficients of the stiffness of the} blanket ( )

, i j

k n (their calculation is described in

paragraph 2.2) as well as the coefficients of the stiffness of the assemblies of bearings and the cylinders that reflect the deformations of the latter. All the elements of the matrix [K(i)_{] are constants.}

The vector F(i) _{of the forces shifting the cylinders} in the i-th approximation is a function of the approximate values of shifts of the assemblies of bearings

( ) ( ) ( )

2, 2, 4 i i i x y x

d d d _{found in the course of the calculation. It}

includes only three constant non-zero pairs of components (the same two components for each shift):

(14).

The components of the vector FG of the

forces of gravity are constants as well.

The system of Equations (13) was formed by using a computer-aided method following the scheme of calculations and is provided in Fig. 3. Various algorithms can be used for its formation; however, we recommend the following algorithm that was developed by us. First, equations of the subsystems of the system (not of the whole system) are formed for separate cylinders, as are the elastic elements simulating the blanket that are artificially separated from one of the cylinders. In such a way, in the example under discussion, the following separate systems of equations of the six subsystems are formed: the equations of cylinder 1 with the elastic elements simulating the blanket that are separated from cylinder 2 (their coefficients of stiffness are ( )

1,i

k n) – these equations describe its shifts in the

direction x1; the equations of cylinder 2 that describe its shifts in the direction x2 with elastic elements separated from cylinder 3 (their coefficients of stiffness are ( )

2,i

k n);

the equations of cylinder 2 without elastic elements that describe its shifts in the direction y2; two analogous systems of equations without elastic elements that separately describe the shifts of cylinder 3 in the directions

x3 and y3; the equations of cylinder 4 that describe its shifts in the direction x4 with the elastic elements separated from cylinder 3 (their coefficients of stiffness are ( )

3,i

k n). In addition, linear algebraic link equations are

formed; they describe the links of the free ends of the elastic elements simulating the blanket with the cylinders artificially separated from them.

After the development of all six systems of the equations of subsystems and the link equations, they are united into a single system (13) using the computer-aided method. Such a method for the formation of the

( )

( )

( ) ( ) ( ) ( )

( )

( )

2 ,2 2 ; 2 ,2 2; 4 ,4 4

i i i i i i

x b x y b y x b x

uporablja posamezne podsisteme valjev, je koristna, ker z njo lahko na podlagi poznanih enaèb podsistemov valjev, preprosto oblikovanih povezovalnih enaèb in mnogih drugih zapletenih enaèb, ustvarimo celoten sistem na naèin, ki lahko vkljuèuje razlièna števila valjev in njihove medsebojne lege. Ta metoda razvoja enaèb je podrobno opisana v virih [8] in [9].

Sedaj predpostavimo, da so vrednosti pomikov sklopov ležajev, kakor tudi komponente vektorja ( )

(

( ) ( ) ( )

)

2, 2, 4 i i i i

x y x d d d

F , znane. Ko rešimo sistem

nelinearnih algebraiènih enaèb (13), ugotovimo, da potekajo absolutni pomiki v nadzornih toèkah A1,...,A4 v smereh osi koordinat x1( )i,...,x( )4i,y2( )i,y3( )i

v i-tem približku. Zdaj lahko ugotovimo, da so spremembe razdalj A1A2, A2A3, A3A4, tj. spremembe

stiskov obloge med nadzornimi toèkami v i-tem približku naslednje:

Absolutne linearne pomike ( )1,1,..., ( )4,4

i i

A x A x

q q _v

toèkah A1,...,A4 in v smereh ustreznih osi koordinat

smo dobili v postopku rešitve sistema (13). Namesto enaèb (15) lahko zapišemo naslednje:

Enaèbe (13) so linearne in pri njihovih rešitvah moramo upoštevati naèelo nalaganja. Zato so enaèbe (16), ki smo jih dobili z rešitvijo enaèb (13), veljavne in koeficienti g(i)_{z ustreznimi indeksi}

bodo ostali nespremenjeni za vse konène vrednosti

( ) ( ) ( )

2, 2, 4 i i i x y x

d d d _, 1 2 3 4 ( ) ( )

, ,..., ,

i i

A A G A A G

x x _{(vkljuèno z nièelnimi}

vrednostmi). To znaèilnost enaèb (16) lahko uporabimo pri izraèunu koeficientov g(i) _in

komponent 1 2 3 4

( ) ( )

, ,..., ,

i i

A A G A A G

x x _{, kakor tudi pri izraèunu}

pomikov ležajev v i-tem približku. To bo izvedeno takole.

1. Izraèun koeficientov g(i) _{(predpostavljamo, da je}

FG = 0):

Koeficiente 1 2 2 2 3 2 3 4 2 ( ) ( ) ( )

, , , , ,

i i i

A A x A A x A A x

g g g

izraèu-namo po rešitvi sistema (13) in enaèb (16), ki so oblikovane na predpostavki, da je ( )2 ( )4 0;

i i y x d =d =

equations by using separate subsystems of the cylinders is useful, because on the basis of known equations of the subsystems of cylinders and the easily formed link equations and many more complicated equations the whole system can be formed in a way that involves various numbers of cylinders and their locations with respect to each other. This method of the development of the equations is described in detail in [8] and [9].

Let us suppose that the values of the shifts of the assemblies of bearings are known and the components of the vector ( )

(

( ) ( ) ( )

)

2, 2, 4 i i i i

x y x d d d

F are

known as well. Then, after having a solution of the system of non-linear algebraic Equations (13), we find that the absolute shifts of the control points are

A1_{( )},...,A4 in the directions of the axes of coordinates_{( ) ( ) ( )}

1i,..., 4i, 2i, 3i

x x y y in the i-th approximation. Then,

the changes of the distances A1A2, A2A3, A3A4, i.e., the compressions of the blanket between the control points in the i-th approximation are found to be:

(15).

Where the absolute rectilinear shifts

( ) ( )

1,1,..., 4,4

i i

A x A x

q q _{of the points A1,...,A4 in the directions} of the relevant axes of the coordinates are found during the solution of the system (13). Instead of Equations (15) we can write the following:

(16).

The equations (13) are linear and the principle of superposition is valid for their solution. Because of this, the equalities (16), found from the solution of the equations (13), are valid and the coefficients

g(i) _{with the relevant indices will not change on any} finite values of ( )2, ( )2, ( )4

i i i x y x

d d d _, 1 2 3 4 ( ) ( )

, ,..., ,

i i

A A G A A G

x x

(including zero values). This feature of the equations (16) is applied to a calculation of the coefficients g(i)

and the components 1 2 3 4

( ) ( )

, ,..., ,

i i

A A G A A G

x x _{and then to a}

calculation of the shifts of the bearings in the i-th approximation. This will be carried out as follows.

1. Calculation of the coefficients g(i) _{(it is considered} that FG = 0):

The coefficients 1 2 2 2 3 2 3 4 2

( ) ( ) ( ) , , , , ,

i i i

A A x A A x A A x

g g g _are

calculated on the basis of solutions of system (13) and the equations (16) obtained on the consideration

( ) ( )

(

( ) ( )

)

( ) ( ) ( )

( ) ( ) ( ) ( )

1 2 1 1 2 2 2 2

2 3 2 2 3 3

3 4 3 3 3 3 4 4

, , , 1 , 1

, , ,

, , 2 , 2 ,

cos sin

cos sin

i i i i

s A A A x A x A y

i i i

s A A A x A x

i i i i

s A A A x A y A x

q q q

q q

q q q

a a

a a

D = - +

D = +

D = +

-( ) ( )

( )

( ) ( ) ( ) ( )

( )

( ) ( )

( ) ( )

( )

( ) ( ) ( ) ( )

( )

( ) ( )

( ) ( )

( )

( ) ( ) ( ) ( )

( )

( ) ( )

1 2 1 2 2 2 1 2 2 2 1 2 4 4 1 2

2 3 2 3 2 2 2 3 2 2 2 3 4 4 2 3

3 4 3 4 2 2 3 4 2 2 3 4 4 4 3 4

, , , , ,

, , , , ,

, , , , ,

i i i i i i i i

s A A A A x x A A y y A A x x A A G

i i i i i i i i

s A A A A x x A A y y A A x x A A G

i i i i i i i i

s A A A A x x A A y y A A x x A A G

x

x

x

g d g d g d

g d g d g d

g d g d g d

D = × - + × + × - +

D = × - + × + × - +

( )

2 1 i x

d = _{. V tem primeru je sistem deformiran z}

dvema enakima posplošenima silama Fx2 = kb,2, ki

vplivata na robove valja 2 in sta edini nenièelni komponenti vektorja F(i)_{. Koeficienti, ki jih}

moramo doloèiti, kar je razvidno iz enaèb (16), za omenjene razmere, bodo enaki vrednostim

( ) ( )

1 2 2 3 , , ,

i i

s A A s A A

D D in ( )

3 4 , i s A A

D .

Koeficiente 1 2 2 2 3 2 3 4 2 ( ) ( ) ( )

, , , , ,

i i i

A A y A A y A A y

g g g

izraèu-namo na naèin, ki je analogen sistemu (13),

( ) ( )

2 4 0

i i x x

d =d = _in ( )i₂ 1

y

d = _{(na sistem vplivata dve}

po-splošeni sili Fy2 = kb,2), koeficiente

1 2 4 2 3 4 3 4 4 ( ) ( ) ( )

, , , , ,

i i i

A A x A A x A A x

g g g _{pa dobimo s predpostavko,}

da je ( )2 ( )2 0 i i x y

d =d = _in ( )i₄ 1

x

d = _{(na sistem vplivata dve}

posplošeni sili Fx4 = kb,4).

2. Komponente 1 2 2 3 3 4 ( ) ( ) ( )

, , , , ,

i i i

A A G A A G A A G

x x x _{, ki se pojavijo}

zaradi sile teže, ki delujejo na valje, dobimo iz enaèb (13) in (16), èe upoštevamo, da je ( )2 ( )2 ( )4 0

i i i

x y x

d =d =d = in FG ¹ 0. Po izraèunu, dobljenem na tej osnovi, bodo komponente enake vrednostim ( ) ( )

1 2 3 4

, ,..., ,

i i

s A A s A A

D D .

3. Ko dobimo koeficiente 1 2 2 3 4 4 ( ) ( )

, ,..., ,

i i

A A x A A x

g g _in

komponente, ki se pojavijo zaradi sile teže, enaèbe (16) uporabimo za doloèitev približnih pomikov sklopov ležajev valjev. Za ta namen približne vrednosti stiskanja gumijastih oblog ( ) ( )

1 2 3 4

, ,..., ,

i i

s A A s A A

D D

na levi strani enaèb (16) nadomestimo z natanènimi potrebnimi vrednostmi stiskanja gumijastih oblog

Ds,A1 A2, Ds,A2 A3, Ds,A3 A4. Približne vrednosti pomikov

sklopov ležajev nato dobimo iz naslednjega sistema enaèb:

2.2 Stopnje izraèuna

Na podlagi ugotovitev iz poglavja 2.1, pomike valjev izraèunamo v naslednjih stopnjah.

1. stopnja (nièti približek, i = 0)

a) Predpostavimo, da so valji absolutno togi. Èe so potrebne vrednosti stiskanja gumijastih oblog

Ds,A1 A2, Ds,A2 A3, Ds,A3 A4 znane in jih vnesemo v

enaèbo (8) namesto Dm,j, bomo dobili nièti približek koeficientov togosti elastiènih elementov obloge ( )0 ( )0 ( )0

1, , 2, , 3,

k n k n k n .

that ( )2 ( )4 0; 1( )2

i i i

y x x

d =d = d = _{. In such a case, the system}

is deformed by two equal generalized forces Fx2 =

kb,2 affecting the ends of cylinder 2, which are the only non-zero components of the vector F(i)_{, and} the coefficients to be found, as can be seen from the equations (16), on the said conditions, will be equal to the values of ( ) ( )

1 2 2 3 , , ,

i i

s A A s A A

D D and ( )

3 4 , i s A A

D .

The coefficients 1 2 2 2 3 2 3 4 2

( ) ( ) ( ) , , , , ,

i i i

A A y A A y A A y

g g g _are

calculated in a analogous way to that in System (13),

( ) ( )

2 4 0

i i x x

d =d = _and ( )i₂ 1

y

d = _{(the system is affected by} two generalized forces Fy2 = kb,2) and the coefficients

1 2 4 2 3 4 3 4 4 ( ) ( ) ( )

, , , , ,

i i i

A A x A A x A A x

g g g _{– on the consideration that}

( ) ( )

2 2 0

i i x y

d =d = _and ( )i₄ 1

x

d = _{(the system is affected by}

two generalized forces Fx4 = kb,4).

2. The components 1 2 2 3 3 4

( ) ( ) ( ) , , , , ,

i i i

A A G A A G A A G

x x x _{caused by}

the force of gravity acting on the cylinders are found from the equations (13) and the equalities (16), taking into account that ( )2 ( )2 ( )4 0

i i i

x y x

d =d =d = _{and FG ¹ 0.}

Based on the mentioned conditions, they will be

equal to the values of ( ) ( )

1 2 3 4

, ,..., ,

i i

s A A s A A

D D .

3. After finding the coefficients 1 2 2 3 4 4 ( ) ( )

, ,..., ,

i i

A A x A A x

g g _and

the components caused by the forces of gravity, the equalities (16) are used to determine the approximate shifts of the assemblies of bearings of the cylinders. For this purpose, the approximate values of the

compressions of the blankets ( ) ( )

1 2 3 4

, ,..., ,

i i

s A A s A A

D D on the

left-hand side of the equalities (16) are replaced with the exact required values of the compression of the blankets Ds,A1 A2, Ds,A2 A3, Ds,A3 A4. Then the approximate values of the shifts of the assemblies of the bearings are found from the following system of equations:

(17).

2.2 The stages of calculation

Following the statements provided in sub-paragraph 3.1, the shifts of the cylinders will be calculated in the following stages.

The Stage 1 (the zero approximation, i = 0)

a) The cylinders are considered absolutely stiff. If the required values of the compression of the blankets

Ds,A1 A2, Ds,A2 A3, Ds,A3 A4 are known and they are written into formula (8) instead of Dm,j, the coefficients of

stiffness ( )0 ( )0 ( )0 1, , 2, , 3,

k n k n k n of the elastic elements of the

blanket are found in the zero approximation.

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 2 2 1 2 2 2 1 2 4 4 1 2 1 2

2 3 2 2 2 3 2 2 2 3 4 4 2 3 2 3

3 4 2 2 3 4 2 2 3 4 4 4 3 4 3 4

( ) ( ) ( ) ( ) , , , , , ( ) ( ) ( ) ( ) , , , , , ( ) ( ) ( ) ( , , , , ,

i i i

i i i i

A A x x A A y y A A x x s A A A A G

i i i

i i i i

A A x x A A y y A A x x s A A A A G

i i i

i i i i

A A x x A A y y A A x x s A A A A G

x x x

g d g d g d

g d g d g d

g d g d g d

- + - = D

-- + - = D

-- + - = D - )

b) Potem ko izraèunamo navedene koeficiente togosti, z uporabo teh koeficientov in drugih znanih parametrov obravnavanega sistema sestavimo matriko koeficientov togosti [K(0)_{] v}

niètem približku.

c) Nato izraèunamo koeficiente 1 2 2 3 4 4 (0) (0)

, ,..., ,

A A x A A x

g g

iz enaèbe (16) in komponente 1 2 (0)

, , A A G

x

2 3 3 4 (0) (0)

, , , A A G A A G

x x _{, ki se pojavijo zaradi sile teže, ki}

deluje na valje.

d) Iz enaèb (17) dobimo vrednosti ( )2 ( )2 ( )4 0_, 0_, 0 x y x d d d _{, nato}

pa sestavimo vektor ( )

(

( ) ( ) ( )

)

2 2 4

0 0_, 0_, 0 x y x d d d

F z

nenièelnimi komponentami (14).

e) Rešimo sistem niètega približka (13), pri katerem upogibe in druge pomike valjev ocenimo v niètem približku:

in dobimo niète približke vrednosti koordinat q(0)_.

2. stopnja (prvi približek, i=1)

a) Ko poznamo vrednosti koordinat q(0)_{, tj. ko}

ocenimo niète približke pomikov valjev, vkljuèno z upogibi, lahko z enaèbo (9) bolj natanèno doloèimo koeficiente togosti elastiènih elementov gumijaste obloge, nato pa tudi bolj natanèno definiramo matriko koeficientov togosti

[K(1)_].

b) Zdaj izraèunamo bolj natanèno definirane koeficiente 1 2 2

(1) , A A x

g _{, … ,} 3 4 4

(1) , A A x

g _{iz enaèb (16) pa}

tudi komponente 1 2 2 3 3 4 (1) (1) (1)

, , , , , A A G A A G A A G

x x x _.

c) Iz enaèb (17) dobimo približne vrednosti pomikov sklopov ležajev ( )2 ( )2 ( )4

1_, 1_, 1 x y x

d d d _{, nato pa sestavimo}

vektor ( )

(

( ) ( ) ( )

)

2 2 4 1 1_, 1_, 1 x y x d d d

F z nenièelnimi

komponentami (14).

d) Sestavimo naslednji sistem enaèb:

Ko ga rešimo, dobimo vrednosti prvega približka koordinat q(1)_.

Naslednje stopnje (i=2, 3, …)

S približnimi vrednostmi koordinat q(i-1)_,

dobljenih na predhodni stopnji, bolj natanèno definiramo koeficiente togosti gumijaste obloge in sestavimo matriko [K(i)_{]. Nato dobimo koeficiente}

1 2 2 3 4 4 ( ) ( )

, ,..., ,

i i

A A x A A x

g g _{iz enaèb (16) in komponente}

1 2 2 3 3 4 ( ) ( ) ( )

, , , , ,

i i i

A A G A A G A A G

x x x _{, iz enaèb (17) pa dobimo}

b) After the calculation of the said coefficients of stiffness is completed, the matrix of the coefficients of stiffness [K(0)_{] in the zero approximation is} formed by using that coefficient and other known parameters of the system under discussion.

c) Then the coefficients 1 2 2 3 4 4

(0) (0)

, ,..., ,

A A x A A x

g g _{from the}

equality (16) and the components

1 2 2 3 3 4 (0) (0) (0)

, , , , , A A G A A G A A G

x x x _{caused by the force of}

gravity acting on the cylinders are calculated. d) The values of ( )2 ( )2 ( )4

0_, 0_, 0 x y x

d d d _{are found from the}

equations (17) and then the vector ( )

(

( ) ( ) ( )

)

2 2 4

0 0_, 0_, 0 x y x d d d F

with non-zero components (14) is formed.

e) The system of zero approximation (13), where bends and other shifts of the cylinders are assessed in the zero approximation, is solved:

(18)

and the values of the coordinates q(0) _{in the zero} approximation are found.

The Stage 2 (the first approximation, i=1)

a) When the values of the coordinates q(0) _{are known,} i.e., after an assessment of the shifts of the cylinders in the zero approximation, including the bends, the coefficients of the stiffness of the elastic elements of the blanket are more closely defined by using formula (9) and then the more closely defined matrix of the coefficients of stiffness [K(1)_{] is formed.} b) The more closely defined coefficients 1 2 2

(1) , A A x

g _,

… , 3 4 4

(1) , A A x

g _{included into the equalities (16) as}

well as the components 1 2 2 3 3 4

(1) (1) (1) , , , , , A A G A A G A A G

x x x _are

then calculated.

c) The approximate values of the shifts of the assemblies of bearings ( )2 ( )2 ( )4

1_, 1_, 1 x y x

d d d _{are found from}

Equations (17), and the vector ( )

(

( ) ( ) ( )

)

2 2 4 1 1_, 1_, 1 x y x d d d F

with non-zero components (14) is then formed. d) The following system of equations is formed:

(19).

After its solution, the values of the coordinates q(1) _{in the first approximation are found.}

Further stages (i=2, 3, …)

By using the approximate values of the coordinates q(i-1) _{found at the previous stage, the} coefficients of the stiffness of the blanket are more closely defined and the matrix [K(i)_{] is formed. Then, the}

coefficients 1 2 2 3 4 4

( ) ( )

, ,..., ,

i i

A A x A A x

g g _{included in the equalities}

(16), and the components 1 2 2 3 3 4

( ) ( ) ( ) , , , , ,

i i i

A A G A A G A A G

x x x _are

( ) ( ) ( )

(

( ) ( ) ( )

)

2 2 4

0 0 0 0_, 0_, 0

x y x G

d d d

é ù = +

ëK ûq F F

( ) ( ) ( )

(

( ) ( ) ( )

)

2 2 4 1 1 1 1_, 1_, 1

x y x G

d d d

é ù = +

found, and the values of the shifts ( )2, ( )2, ( )4 i i i x y x d d d _are

calculated from the equations (17). Then, after the formation of the vector ( )

(

( ) ( ) ( )

)

2, 2, 4 i i i i

x y x d d d

F , systems of

equations, where i = 2, 3, 4 and so on, are formed. In the course of finding a solution to each of these systems, the values of the coordinates q(i) _{are more closely defined.} The calculation at these stages is continued until the differences between the coordinates q(i) _and

q(i-1)_{, the components of the vectors, the shifts of the} assemblies of cylinder bearings and other values found during the two last stages become less than 1%. In such a case the calculation is then stopped.

From the obtained results the necessary values of the shifts of the cylinder bearings as well as the regularities of the distribution of deformations of the cylinders and the pressure between the cylinders that are pressed against each other via blankets are found. By using these methods in practice we found that sufficient accuracy (when the error is less than 1%) is achieved when i = 4.5.

3 THE EXAMPLE

By using the method described herein, the bends of the cylinders of the “Compacta 213” printing station, made in Germany, and the distribution of the pressure along their working surfaces were investigated. The shifts of the solid and the pipe-type cylinders were investigated. The parameters of the cylinders were as follows (the same for all cylinders): L =1.384 m; l1=1.04m; l2=1.00m; l3=0.96m; Dd=0.2m; kb,1

=kb,2=kb,3 =kb,4=1.7×106N/m; E =500MPa; h=1.5; a1

= 30°; a2= 50°; g= 35°; the diameter dd of cylinder

holes was varied, and each cylinder was divided into 13 finite elements (see Fig. 3).

After the completion of the investigation it was found that the relative bends of the blanket and the plate cylinders that were pressed against each other can be up to 24 mm and cause 14–31% changes in the pressure along the working surfaces of the cylinders. This may noticeably deteriorate the quality of the prints. It was shown that it is not useful to use pipe-type cylinders with thin walls (the cylinders of the “Compacta 213” printing press are solid). In order to equalize the distribution of the pressure, the working surfaces of the cylinders can be made convex or concave, but not cylindrical. For this purpose, by using the special programme, changes of the diameters of the cylinders are chosen in such way so as to ensure a rectilinear surface of the contact of the cylinders. In such a case, the pressure along the contact bands of the cylinders remains constant.

vrednosti pomikov ( )2, ( )2, ( )4 i i i x y x

d d d _{. Po oblikovanju}

vektorja ( )

(

( ) ( ) ( )

)

2, 2, 4 i i i i

x y x d d d

F , dobimo sisteme enaèb,

kjer je i = 2, 3, 4 itn. V postopku iskanja rešitve vsakega od sistemov bolj natanèno definiramo vrednosti koordinat q(i)_.

Na teh stopnjah z izraèunom nadaljujemo, dokler razlike med koordinatama q(i) _{in q}(i-1)_,

komponentami vektorjev, pomiki sklopov ležajev valjev in drugimi vrednostmi, ki smo jih dobili med potekom zadnjih dveh stopenj, ne postanejo manjše od 1%. Ko se to zgodi, prenehamo z izraèunavanjem.

Iz dobljenih rezultatov dobimo potrebne vrednosti pomikov ležajev valjev pa tudi zakonitost razporeditve deformacij valjev ter tlaka med valjema, ki pritiskata drug ob drugega preko gumijaste obloge.

Z uporabo teh metod v praksi smo ugotovili, da dosežemo zadostno natanènost rezultatov (ko napaka znaša manj kot 1%), ko je i = 4,5.

3 PRIMER

Z uporabo tu opisane metode smo raziskali upogibe valjev tiskarskega stroja “Compacta 213”, izdelanega v Nemèiji, in razporeditev tlaka vzdolž njihovih delovnih površin. Obdelali smo pomike polnih valjev in cevastih valjev. Parametri valjev so bili naslednji (isti za vse valje) L =1,384 m; l1=1,04m;

l2=1,00m; l3=0,96m; Dd=0,2m; kb,1=kb,2=kb,3 =kb,4

=1,7×106_{N/m; E =500MPa; h=1,5; a}

1= 30°; a2= 50°;

g= 35°; spreminjali smo premer odprtin valjev dd, vsak valj pa smo razdelili na 13 konènih elementov (sl. 3).

Po opravljeni raziskavi smo ugotovili, da relativni upogibi valjev z gumijasto oblogo in valjev s plošèo, ki pritiskajo drug ob drugega, lahko znašajo do 24 mm in povzroèijo 14 do 31 odstotkov sprememb v tlaku vzdolž delovnih površin valjev. To lahko vidno poslabša kakovost tiska. Pokazali smo, da ni praktièno uporabljati cevaste valje s tankimi stenami (valji tiskarskega stroja “Compacta 213” so polni).

24 22 20 18 16 14 12 10

100 80

60 40 20 0

Re

lativ

e be

nd

,

Thickness of walls, m m

relativni upogib relative bend

debelina sten thickness of walls

Sl. 4. Najveèji relativni upogib med delovnima površinama valjev z gumijasto oblogo 2, 3 pri razliènih debelinah sten votlih valjev 1, 2, 3, 4 (so enake za vse štiri valje)

Fig. 4. The maximum relative bend between the working surfaces of the blanket cylinders 2, 3 on the various widths of the walls of the hollow cylinders 1, 2, 3, 4 (the same for all four cylinders)

0 200 400 600 800 1000 1200

0,92 0,94 0,96 0,98 1,00 1,02

Pr

es

su

re

p m , 1

MP

a

Distance l 2, mm

Distance a l m , mm

0 200 400 600 800 1000 2

0

1.

6 9

0.

4 9

0.

8 9

0.

2 9

0.

0 0

1.

3 4

5 6

1 2

0 200 400 600 800 1000 1200

1,0 1,1 1,2 1,3 1,4 1,5 1,6

Pr

ess

ure

p m , 2

MPa

Distance l 2,mm

Distance b l m , mm

0 200 400 600 800 1000 6

1.

5

1.

4

1.

3

1.

2

1.

1

1.

0

1.

1 2

3 ₄

5 6

0 200 400 600 800 1000 1200

0,84 0,88 0,92 0,96 1,00

0

1.

6 9

0.

2 9

0.

88

0.

2 8

0.

1 2

3 4 5

6

0 200 400 600 800 1000

Pr

essur

e

p m

, 3

MPa

Distance l ₂ , mm

Distance _c l _m , mm

tlak pressure _{tlak pressure}

tlak pressure

razdalja distance

razdalja distance

razdalja distance

Sl. 5. Porazdelitev tlaka pm,j vzdolž delovnih površin valjev, ki pritiskajo drug ob drugega, za primer ocenjenih upogibov valjev: a – med valjem s plošèo 1 in valjem z gumijasto oblogo 2; b – med valjema z gumijasto oblogo 2 in 3; med valjem z gumijasto oblogo 3 in valjem s plošèo 4; 1 – polni valji; 2 – votli

valji z debelino sten 50 mm; 3 – 40 mm, 4 – 30 mm, 5 – 20 mm, 6 – 10 mm

Fig. 5. The distribution of the pressure pm,j along the working surfaces of cylinders that are pressed against each other on an assessment of the bends of the cylinders: a – between the plate cylinder 1 and

the blanket cylinder 2; b – between the blanket cylinders 2 and 3; between the blanket cylinder 3 and the plate cylinder 4; 1 – solid cylinders; 2 – hollow cylinders with the width of the walls 50 mm; 3 – 40